The end members, Bi and Bi2Te3, of the infinitely adaptive (Bi2)m(Bi2Te3)n series of compounds have not only been revealed to be topological insulators under the appropriate conditions, but have also been shown to be superconductors under pressure, suggesting the potential for bulk superconductor-topological- insulator interfaces and associated quantum computing applications. Herein, we report the pressure-dependent evolution of the structure and electrical transport of the nominal semimetal Bi4Te3, a member of the (Bi2)m(Bi2Te3)n series. Under pressure, Bi4Te3 undergoes several structural phase transformations, ultimately yielding a metallic body-centered-cubic structure exhibiting superconductivity with a maximum Tc=8.4 K at 16.2 GPa. The occurrence of structure-dependent superconductivity in Bi4Te 3 is remarkably similar to the end members of the (Bi 2)m(Bi2Te3)n series, intimating a convergence to high-pressure universal behavior that may expose the subtle variations that lead to the topological insulating and superconducting states in these systems. © 2011 American Physical Society.